Final answer:
The equations of the asymptotes for the given hyperbola are x = 0 (the y-axis) and x = 0 (the x-axis).
Step-by-step explanation:
The equations of the asymptotes of a hyperbola can be found using the formula y = mx + b, where m is the slope of the asymptote and b is the y-intercept. In the case of a hyperbola centered at the origin, the slopes of the asymptotes are given by the ratio of the y-coordinate of the focus to the x-coordinate of the vertex: m = ±(y-coordinate of the focus) / (x-coordinate of the vertex).
For the given hyperbola with a vertex at (0,-40) and a focus at (0, 41), the slope of the asymptotes is:
m = ±41 / 0 = ±∞
Since the slopes are undefined (vertical lines), the equations of the asymptotes are of the form x = c, where c is a constant. Therefore, the equations of the asymptotes for this hyperbola are x = 0 (the y-axis) and x = 0 (the x-axis).